Mitigating polarization dependent loss (PDL) by transforming frequency components to a ball

ABSTRACT

An apparatus for mitigating polarization dependent loss (PDL) in an optical signal-to-noise ratio (OSNR) of a modulated optical signal is disclosed. The apparatus may comprise a spectrum analyzer to measure an optical power spectrum of a modulated optical signal. The apparatus may also comprise a measuring unit to select a first portion of the modulated optical signal and a second portion of the modulated optical signal, where each of the first and second portions of the modulated optical signals may include an independent noise distribution indicative of PDL, and measure a time-varying parameter of the first and second portions. The apparatus may also include a signal processor to PDL in an OSNR by transforming any elliptical polarization associated with the independent noise distribution into a ball polarization, determining a correlation between time-varying parameters of the first and second portions, and calculating a PDL mitigated OSNR.

RELATED APPLICATIONS

The present application is a Continuation of commonly assigned andco-pending U.S. patent application Ser. No. 16/277,592, filed Feb. 15,2019, which is related to U.S. application Ser. No. 15/787,515 toHeismann, entitled “Determining In-Band Optical Signal-to-Noise Ratio inPolarization-Multiplexed Optical Signals Using Signal Correlations,”filed Oct. 18, 2017, which is incorporated herein by reference in itsentirety.

TECHNICAL FIELD

This patent application relates generally to telecommunicationsnetworks, and more specifically, to systems and methods for mitigatingpolarization dependent loss (PDL) by transforming frequency componentsto a ball.

BACKGROUND

Determination of optical signal-to-noise ratio and other signal qualityparameters is important in telecommunications. Quality of modulatedoptical signals transmitted in long-distance fiberoptic communicationsystems is frequently characterized by optical signal-to-noise ratio(OSNR), which defines a ratio of the total optical power of the digitalinformation signal to optical noise added to the signal by opticalamplifiers. In communication systems with only a few widely-spacedwavelength-multiplexed signals, OSNR may be readily determined byspectral analysis of a transmitted noisy signal and the optical noisefloor on either side of the signal spectrum.

In modern optical communication systems with dense wavelength-divisionmultiplexing (DWDM), various transmitted optical signals are so closelyspaced in optical frequency that it becomes difficult to measure opticalnoise floor between adjacent signal spectra. This is of a particularconcern for communication systems transmitting optical signals at bitrates of 100 Gb/s over 50-GHz wide wavelength channels. In thesesystems, optical noise floor within the spectral bandwidth of the signalneeds to be measured to determine the signal's OSNR. Such measurementsare commonly referred to as in-band OSNR measurements. Furthermore, itis frequently required that these in-band OSNR measurements areperformed while the communication system is in service, e.g., that thenoise floor within the signal's bandwidth is determined while theoptical information signal is transmitted.

Several methods have been disclosed to measure an in-band OSNR inpresence of transmitted signals. For conventional single-polarizedoptical information signals (e.g., for 10 Gb/s NRZ-OOK signals), apolarization-nulling technique can be used, which substantially removesthe polarized signal from the received noisy signal, thus revealing thefloor of an unpolarized optical noise in the spectral bandwidth of thesignal.

Modern optical information signals, however, are frequently composed oftwo mutually orthogonally polarized optical carriers at a same opticalfrequency. Such carriers are typically independently modulated withdigital information data. This polarization multiplexing technique isfrequently used in long-distance communication systems to transmit 50Gb/s BPSK, 100 Gb/s QPSK, or 200 Gb/s 16-QAM signals over 50-GHz wideDWDM channels. In polarization-multiplexed (PM) signals, in-band OSNRcannot be determined by means of the above-referencedpolarization-nulling technique because the two orthogonally polarizedoptical carriers cannot be simultaneously removed from the noisy opticalsignal without also extinguishing the optical noise.

While several methods have been proposed to measure in-band OSNR inpolarization-multiplexed signals, they generally only work with opticalsignals of a predetermined bit-rate, modulation format, and/or signalwaveform. Consequently, these conventional methods may be suitable formonitoring of in-band OSNR at certain points in a communication system,e.g., by means of built-in monitoring equipment, but may be difficult touse as a general test and measurement procedure. Furthermore, some ofthese methods are not suitable for determining in-band OSNR in signalssubstantially distorted by chromatic dispersion (CD) orpolarization-mode dispersion (PMD). In addition, there is currently nosolution to obtain a good maximum correlation whenpolarization-dependent loss (PDL) is present on the fiber link.

As a result, techniques that mitigate polarization-dependent loss (PDL)may be helpful when determining in-band OSNR in signals substantiallydistorted by chromatic dispersion (CD) or polarization-mode dispersion(PMD) and overcome shortcomings of conventional technologies.

BRIEF DESCRIPTION OF DRAWINGS

Features of the present disclosure are illustrated by way of example andnot limited in the following Figure(s), in which like numerals indicatelike elements, in which:

FIG. 1 illustrates a plot of an optical power spectrum of a noisy 100Gb/s polarization-multiplexed RZ-QPSK signal, showing spectralcomponents used for measuring the correlation coefficients at two spacedapart optical frequencies;

FIG. 2 illustrates a plot of a normalized amplitude spectral correlationdensity function (SCDF) evaluated at two spaced apart opticalfrequencies and of the OSNR estimated form the SCDF versus “true”in-band OSNR in a 100 Gb/s polarization-multiplexed RZ-QPSK signal;

FIG. 3 illustrates a plot of an optical power spectrum of a noisy 100Gb/s polarization-multiplexed RZ-QPSK signal, showing an example of thespectral components used for correlation measurements, in which thecenter frequency ƒ is offset from the carrier frequency ƒ_(c) of thesignal;

FIG. 4 illustrates a plot of a normalized amplitude SCDF and estimatedOSNR versus frequency offset ƒ−ƒ_(c) for a 100 Gb/spolarization-multiplexed RZ-QPSK signal;

FIG. 5 illustrates a plot of optical power variations in two spectralcomponents of a substantially noiseless 100 Gb/s PM-QPSK signal (>40 dBOSNR), measured at two spaced apart optical frequencies, with an opticalbandwidth of 200 MHz;

FIG. 6 illustrates a plot of optical power variations in two spectralcomponents of a noisy 100 Gb/s PM-QPSK signal having an OSNR of only 12dB OSNR, measured at two spaced apart optical frequencies, with anoptical bandwidth of 200 MHz;

FIG. 7 illustrates a plot of optical power variations of two spectralcomponents of a noiseless 100 Gb/s PM-QPSK signal distorted by chromaticdispersion with 50,000 ps/nm GVD,

FIG. 8 illustrates a plot of the normalized power SCDF at two spacedapart optical frequencies centered around ƒ_(c), and the estimated OSNRcalculated for a 100 Gb/s PM RZ-QPSK signal;

FIG. 9 illustrates a plot of the normalized power SCDF and the estimatedOSNR versus frequency offset ƒ−ƒ_(c) calculated for a 100 Gb/s PMRZ-QPSK signal;

FIG. 10 is a schematic illustration of an apparatus for measuringamplitude correlations in the spectrum of a modulated signal, using twoparallel coherent receiver channels with phase and polarizationdiversity and subsequent digital signal processing;

FIG. 11 illustrates a plot of OSNR estimated from amplitude SCDF in acoherently received 100 Gb/s PM RZ-QPSK signal versus reference OSNR,calculated for a receiver with 5th-order Butterworth electric low-passfilter having four different bandwidths;

FIG. 12 illustrates a plot of a normalized amplitude and power SCDFs ina 100 Gb/s PM RZ-QPSK signal versus frequency separation of localoscillator (LO) lasers, displayed versus frequency deviation. The solidcurves assume an electrical receiver with 5^(th)-order Butterworthlow-pass filter and the dashed curve a receiver with 5^(th)-order Bessellow-pass filter, both having a 3-dB bandwidth of 40 MHz;

FIG. 13 is a schematic illustration of an apparatus for measuring theoptical power spectrum of the noisy signal, using a coherent receiverwith continuously tunable LO laser;

FIG. 14 is a schematic illustration of an apparatus for measuringintensity correlations in the spectrum of a modulated signal, using twoparallel coherent receivers with phase and polarization diversity andfast digital signal processing;

FIG. 15 is a schematic illustration of another apparatus embodiment formeasuring amplitude or intensity correlations in the optical spectrum ofa modulated signal, using two parallel heterodyne receivers withpolarization diversity and digital down-conversion with phase diversityin a digital signal processor;

FIG. 16 is a schematic illustration of another apparatus embodiment formeasuring amplitude or intensity correlations in the optical spectrum ofa modulated signal, using two parallel coherent receivers and a singlelaser in combination with an optical frequency shifter to generate thetwo optical local oscillator signals spaced apart in frequency;

FIGS. 17A-17C are schematic illustrations of three exemplary embodimentsof optical frequency shifters for generating one or two frequencyshifted optical signals from a single-frequency optical input signal;

FIG. 18 is a schematic illustration of an apparatus embodiment formeasuring intensity correlations in the optical spectrum of a modulatedsignal using two narrowband optical band-pass filters and non-coherentphoto-receivers; and

FIG. 19 illustrates a flow chart of a method for mitigating polarizationdependent loss (PDL) in an optical signal-to-noise ratio (OSNR)measurement, according to an example.

DETAILED DESCRIPTION

For simplicity and illustrative purposes, the present disclosure isdescribed by referring mainly to examples and embodiments thereof. Inthe following description, numerous specific details are set forth inorder to provide a thorough understanding of the present disclosure. Itwill be readily apparent, however, that the present disclosure may bepracticed without limitation to these specific details. In otherinstances, some methods and structures readily understood by one ofordinary skill in the art have not been described in detail so as not tounnecessarily obscure the present disclosure. As used herein, the terms“a” and “an” are intended to denote at least one of a particularelement, the term “includes” means includes but not limited to, the term“including” means including but not limited to, and the term “based on”means based at least in part on.

As described above, quality of modulated optical signals transmitted inlong-distance fiberoptic communication systems is frequentlycharacterized by optical signal-to-noise ratio (OSNR), which defines aratio of the total optical power of the digital information signal tooptical noise added to the signal by optical amplifiers. While severalmethods have been disclosed to measure in-band OSNR inpolarization-multiplexed signals, they generally only work with opticalsignals of a predetermined bit-rate, modulation format, and/or signalwaveform. Consequently, these methods may be suitable for monitoring ofin-band OSNR at certain points in a communication system, e.g., by meansof built-in monitoring equipment. However, these tend to be difficult touse as a general test and measurement procedure. Furthermore, some ofthese methods are not suitable for determining in-band OSNR in signalssubstantially distorted by chromatic dispersion (CD) orpolarization-mode dispersion (PMD). In addition, there is currently nosolution to determine a maximum correlation when polarization-dependentloss (PDL) is present on the fiber link.

By way of example, a method for in-band OSNR measurements has beenproposed, but such a method generally only works with binary PSK and ASKsignals. This method does not work with 100 Gb/s PM-QPSK or 200 Gb/sPM-16-QAM signals.

Other methods for in-band OSNR monitoring of polarization-multiplexedsignals have been proposed but they are generally based on coherentdetection with high-speed receivers and subsequent digital signalprocessing. These methods typically operate at a predetermined bit-rate.For example, it may be known that such methods may determines thein-band OSNR from the spread of the four polarization states throughwhich an optical PM QPSK signal cycles rapidly. Clearly, such high-speedpolarization analysis requires prior knowledge of the modulation formatand the bit-rate of the transmitted signal and, furthermore, is verysensitive to signal distortions caused by chromatic dispersion (CD) andpolarization mode dispersion (PMD).

For applications in long-distance communication systems, it may beadvantageous to remove CD- and PMD-induced signal distortions prior todetermining OSNR. Compensation of signal distortions introduced by CDand PMD may be accomplished electronically in a high-speed digitalsignal. Digital compensation, however, requires use of high-speedanalog-to-digital convertors (ADCs), which usually have only arelatively small dynamic range (typically less than 16 dB), thuslimiting the OSNR measurement range. In-band OSNR measurement methodsemploying error vector magnitude (EVM) analysis of the received signalafter electronic compensation of CD and PMD have been proposed and used;however, EVM analysis intrinsically requires foreknowledge of theparticular modulation format of the optical signal.

Another method for OSNR monitoring may be based on RF spectral analysisof low-speed intensity variations of polarization-multiplexed signals.However, this method is very sensitive to variations in signal'swaveform. Hence, may require not only foreknowledge of the modulationformat and bit-rate of the analyzed optical signal, but also carefulcalibration with a noiseless signal.

A method for in-band OSNR measurements using conventional spectralanalysis of the optical signal power has been proposed as well.Unfortunately, this method appears to only work with signals whoseoptical spectrum is substantially narrower than the spectral width ofthe DWDM channel. In other words, this method typically works with 40Gb/s PM NRZ-QPSK signals transmitted through 50-GHz wide DWDM channels,but not with 100 Gb/s PM RZ-QPSK signals transmitted through 50-GHz wideDWDM channels.

Yet another method for in-band OSNR measurements inpolarization-multiplexed signals has been disclosed by W. Grupp inEuropean Patent EP 2,393,223 “In-band SNR measurement based on spectralcorrelation,” issued Dec. 7, 2011. Other methods for in-band OSNRmeasurements have been proposed that rely on determination of in-bandOSNR from measurements of the cyclic autocorrelation function of thesignal amplitude, which is achieved by calculating noiseless signalpower from correlations between spectral components of the Fouriertransform of the cyclic autocorrelation function. The cyclicautocorrelation function of the signal's amplitude may be measured, forexample, by means of two parallel coherent receivers employing a commonpulsed local oscillator laser. Again, this method requires foreknowledgeof the modulation format and bit-rate of the optical signal, as well ascareful calibration of the apparatus with a noiseless signal. Inaddition, the method is very sensitive to signal distortions introducedby CD and/or PMD.

In addition to these promising but ultimately deficient methods andtechniques, there is currently no solution to obtain a reliable maximumcorrelation when polarization-dependent loss (PDL) is present on thefiber link. As a result, techniques that mitigate polarization-dependentloss (PDL) when determining in-band OSNR in signals substantiallydistorted by chromatic dispersion (CD) or polarization-mode dispersion(PMD) are disclosed herein.

Amplitude and phase of digitally modulated optical signals, such asQPSK- and 16-QAM-modulated signals, may vary pseudo-randomly with time.These pseudo-random amplitude and phase variations may be difficult todistinguish from random amplitude and phase variations of optical ASEnoise generated by optical amplifiers, in particular if the waveform ofthe modulated signal is substantially distorted by large amounts ofchromatic dispersion or polarization-mode dispersion in the fiber link.However, an autocorrelation function of digitally modulated signals maybe periodic in time, because the transmitted symbols are assignedpredetermined and substantially equal time intervals, whereas theautocorrelation function of random ASE noise does not exhibit suchperiodicity. The periodicity of the autocorrelation function ofdigitally modulated signals may be manifested in a signal's opticalfrequency spectrum, which may exhibit strong correlations betweentime-varying amplitudes, and also between time-varying intensities andoptical power levels, of certain pairs of spaced apart spectralcomponents, whereas such correlations may not exist in the opticalspectrum of random ASE noise. It may be possible, therefore, todetermine a relative amount of random ASE noise in a transmittedmodulated signal by measuring correlations between the aforementionedspaced apart spectral components, and by subsequently comparing themeasured correlations to corresponding correlations in a noiselesssignal spectrum. Once the relative amount of ASE noise in a transmittedsignal is determined, the in-band OSNR of the noisy signal may becalculated.

Correlations between various spectral components of a digitallymodulated signal may be described by a spectral correlation densityfunction (SCDF), S_(x) ^(α)(ƒ), which may be defined as the Fouriertransformation of the cyclic autocorrelation function, R_(x) ^(α)(τ), ofthe optical signal amplitude x(t), e.g.,

${S_{x}^{\alpha}(f)} \equiv {\int\limits_{- \infty}^{\infty}{{{R_{x}^{\alpha}(\tau)} \cdot {\exp\left( {{- j}\; 2\;\pi\; f\;\tau} \right)}}d\;\tau}}$wherein R_(x) ^(α)(τ) may represent cyclic auto-correlation functiongiven byR _(x) ^(α)(τ)=

x(t+τ/2)·x*(t−τ/2)·exp(−j2παt)

,and x(t) may be a time-varying complex two-dimensional Jones vector,which describes amplitude and phase variations of the two polarizationcomponents of the modulated signal as a function of time t. The brackets< > may denote time averaging over a time period that is substantiallylonger than the symbol period T_(symbol) of the digital modulation. Moregenerally, the averaging period, and accordingly the measurements ofspectral components amplitudes, phases, and/or optical powerlevels/intensities, may be sufficiently long to ensure a predeterminedlevel of fidelity of computed correlations.

Alternatively, the SCDF may be expressed as a correlation function ofthe time-varying amplitudes of the spectral components of the modulatedsignal, e.g., asS _(x) ^(α)(ƒ)=

X _(T)(t,ƒ+α/2)·X _(T)*(t,ƒ−α/2)

wherein

X_(T)(t, ν) = ∫_(t − T/2)^(t + T/2)x(u) ⋅ exp (−j 2 π vu)duand T may be an integration time with T>>T_(symbol). The brackets in theabove expression may denote averaging over a time period substantiallylonger than the integration time T. For methods disclosed herein, it maybe advantageous to define a normalized SCDF of the spectralcorrelations, e.g.,

${{{\overset{\hat{}}{S}}_{x}^{\alpha}(f)} = \frac{\left\langle {X_{T},{\left( {t,{f + {\alpha/2}}} \right) \cdot {X_{T}^{*}\left( {t,{f - {\alpha/2}}} \right)}}} \right\rangle}{\sqrt{\left\langle {{X_{T}\left( {t,{f + {\alpha/2}}} \right)}}^{2} \right\}}\sqrt{\left\langle \left| {X_{T}\left( {t,{f - {\alpha/2}}} \right)} \right|^{2} \right\rangle}}},$which may be similar to the un-balanced correlation coefficient used instatistical analysis, and which has the property −1Ŝ_(x) ^(α)(ƒ)≤1 forall values of ƒ and α.

It should be appreciated that noiseless and otherwise undistortedoptical signals encoded with binary amplitude-shift keying (ASK), binaryphase-shift keying (BPSK), ordinary quaternary phase-shift keying(QPSK), and 16-quadrature-amplitude modulation (16-QAM) may exhibitŜ_(x) ^(α)(ƒ)=1 when α=α₀=1/T_(symbol) and for all frequencies ƒ withinthe range −α/2<ƒ<−α/2. In addition, it should be appreciated that Ŝ_(x)^(α)(ƒ)=1 when α=2/T_(symbol). However, it may be important to note thatfor optical signals encoded with staggered QPSK modulation, alsoreferred to as “offset QPSK”, Ŝ_(x) ^(α)(ƒ)≈0 when α=1/T_(symbol). Forstaggered QPSK Ŝ_(x) ^(α)(ƒ)=1 only when α=2/T_(symbol).

In contrast to modulated optical signals, optical ASE noise may be arandom Gaussian process and, therefore, does not exhibit any significantcorrelation between its spectral components. Therefore, when random ASEnoise is added to a modulated optical signal, the normalized SCDF may besmaller than unity, e.g., S_(x) ^(α)(ƒ)<1, as described below in moredetail. For example, when one allows n(t) to denote the Jones vector ofthe phase and amplitude of random ASE noise added to the transmittedsignal, then the Jones vector {tilde over (X)}_(T)(t,v) of the spectralcomponent of the noisy signal at frequency v may the sum of thenoiseless Jones vector X_(T)(t,v), defined above, and the correspondingJones vector of the ASE noise N_(T)(t,v), e.g.,

${{\overset{˜}{X}}_{T}\left( {t,v} \right)} = {{\int\limits_{t - {T/2}}^{t + {T/2}}{{\left( {{x(u)} + {n(u)}} \right) \cdot {\exp\left( {{- j}\; 2\;\pi\;{vu}} \right)}}{du}}} \equiv {{X_{T}\left( {t,v} \right)} + {{N_{T}\left( {t,v} \right)}.}}}$Consequently, the SCDF of a noisy modulated signal may be expressed as

$\begin{matrix}{{S_{x}^{\alpha}(f)} = \left\langle {{{\overset{\sim}{X}}_{T}\left( {t,{f + {\alpha/2}}} \right)} \cdot {{\overset{\sim}{X}}_{T}^{*}\left( {t,{f - {\alpha/2}}} \right)}} \right\rangle} \\{= {\left\langle {{X_{T}\left( {t,{f + {\alpha/2}}} \right)} \cdot {X_{T}^{*}\left( {t,{f - {\alpha/2}}} \right)}} \right\rangle +}} \\{\left\langle {{N_{T}\left( {t,{f + {\alpha/2}}} \right)} \cdot {N_{T}^{*}\left( {t,{f - {\alpha/2}}} \right)}} \right\rangle}\end{matrix}$wherein the second term on the right side of the equation may vanisheswhen |α|>0. It should be appreciated that

X _(T)(t,ƒ+α/2)·N _(T)*(t,ƒ−α/2)

=

X _(T)(t,ƒ+α/2)·N _(T)*(t,ƒ−α/2)

≡0,because amplitudes of random noise and modulated signal may beuncorrelated. Thus, the normalized SCDF for α>0, e.g. forα=1/T_(symbol), may be represented by

${{\overset{\hat{}}{S}}_{x}^{\alpha}(f)} = \frac{\left\langle {X_{T},{\left( {t,{f + {\alpha/2}}} \right) \cdot {X_{T}^{*}\left( {t,{f - {\alpha/2}}} \right)}}} \right\rangle}{\begin{matrix}\sqrt{\left\langle \left| {X_{T}\left( {t,{f + {\alpha/2}}} \right)} \middle| {}_{2}{+ \left| {N_{T}\left( {t,{f + {\alpha/2}}} \right)} \right|^{2}} \right. \right\rangle} \\\sqrt{\left\langle \left| {X_{T}\left( {t,{f - {\alpha/2}}} \right)} \middle| {}_{2}{+ \left| {N_{T}\left( {t,{f - {\alpha/2}}} \right)} \right|^{2}} \right. \right\rangle}\end{matrix}}$

Assuming that the power spectrum of the modulated signal is symmetricabout its carrier frequency ƒ_(c), which is generally the case for theaforementioned modulation formats, so that

X _(T)(t,ƒ _(c)+α/2)|²

=

X _(T)(t,ƒ _(c)−α/2|)²

≡

P _(S)(ƒ_(c)±α/2)

,and further assuming that the power spectrum of the random ASE also issubstantially symmetric about ƒ_(c), which is frequently the case, sothat

|N _(T)(t,ƒ+α/2)|²

=

|N _(T)(t,ƒ−α/2)|²

≡

P _(N)(ƒ_(c)±α/2)

,then the normalized SCDF at difference frequency α₀1/T_(symbol) may beexpressed as

${{{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}\left( f_{c} \right)} = \frac{\left\langle {P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle}{\left\langle {P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle + \left\langle {P_{N}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle}},$which, after rearrangement, may yield a signal-to-noise ratio, SNR, ofthe spectral components at the two optical frequencies ƒ_(c)−α₀/2 andƒ_(c)+α₀/2, represented as

${{SNR}\mspace{11mu}{\left( {f_{c},\alpha_{0}} \right) = {\frac{\left\langle {P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle}{\left\langle {P_{N}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle} = \frac{{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}\left( f_{c} \right)}{1 - {{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}\left( f_{c} \right)}}}}},$and, similarly, the ratio of the total signal and noise power to thenoise power at ƒ_(c)±α₀/2 as

${\frac{\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + {P_{N}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}} \right\rangle}{\left\langle {P_{N}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle} = \frac{1}{1 - {{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}\left( f_{c} \right)}}}.$

It should be appreciated that the OSNR of a transmitted signal may bedefined as a ratio of the total signal power over the total noise powerin an optical bandwidth B_(noise) (usually equal to 0.1 nm, c.f. IEC61280-2-9 “Digital systems—Optical signal-to-noise ratio measurementsfor dense wavelength-division multiplexed systems”) as

${{{OSNR} \equiv \frac{\sum\limits_{i}{\left\langle {P_{S}\left( f_{i} \right)} \right\rangle B_{meas}}}{\left\langle P_{N} \right\rangle B_{noise}}} = {\frac{\sum\limits_{i}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle B_{meas}}}{\left\langle P_{N} \right\rangle B_{noise}} - {\sum\limits_{i}\frac{B_{meas}}{B_{noise}}}}},$wherein the summation may extend over all frequency components ƒ_(i)within the bandwidth of the signal, and B_(meas) denotes the measurementbandwidth of each power measurement

P_(S)(ƒ_(i))

and

P_(N)(ƒ_(i))

. It should appreciated that the above definition of OSNR assumes thatthe spectrum of the random ASE noise is substantially flat within thebandwidth of the modulated signal, so that the average noise powerdensity is identical at all frequencies, e.g.,

P_(N)(ƒ_(i))

=

P_(N)(ƒ_(j))

≡

P_(N)

for all frequencies ƒ_(i)≠ƒ_(i) within the bandwidth of the modulatedoptical signal. The in-band OSNR of a noisy signal may thus bedetermined from a measurement of the power spectrum of the transmittednoisy signal, i.e. of

P_(S)(ƒ)

+

P_(N)(ƒ)

, and an additional measurement of the average noise power,

P_(N)(ƒ)

, within a bandwidth of the modulated signal.

Whereas conventional single-polarized signals

P_(N)

may be directly measured by blocking the polarized signal with aproperly oriented polarization filter, such measurements may not beperformed with polarization-multiplexed signals. As disclosed herein,

P_(N)

may be obtained from the spectral correlation of the complex Jonesvectors of the signal amplitudes at optical frequencies ƒ_(c)−α₀/2 andƒ_(c)+α₀/2, as described above, and may be expressed as a fraction ofthe noisy signal power at these frequencies, e.g., as

P _(N)

=

P _(S)(ƒ_(c)±α₀/2)+P _(N)

└1−Ŝ _(x) ^(α) ⁰ (ƒ_(c))┘.It should be appreciated that this procedure may be applied topolarization-multiplexed signals, as well as to single-polarizedsignals. Substituting the above relation into the equation for OSNR mayimmediately yield desired in-band OSNR of the optical signal,

$\begin{matrix}{{OSNR} = {{\frac{\sum\limits_{i}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle B_{meas}}}{\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle B_{noise}} \cdot \frac{1}{1 - {{\overset{\hat{}}{S}}_{X}^{\alpha_{0}}\left( f_{c} \right)}}} - {\sum\limits_{i}\frac{B_{meas}}{B_{noise}}}}} \\{= \frac{\sum\limits_{i}{\left\{ {\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle - {\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle\left\lbrack {1 - {{\overset{\hat{}}{S}}_{X}^{\alpha_{0}}\left( f_{c} \right)}} \right\rbrack}} \right\} B_{meas}}}{{\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle\left\lbrack {1 - {{\overset{\hat{}}{S}}_{X}^{\alpha_{0}}\left( f_{c} \right)}} \right\rbrack}B_{noise}}}\end{matrix}$wherein all quantities may be known from the two measurements describedabove.

Therefore, the in-band OSNR of a transmitted noisy signal may bedetermined from a measurement of the spectral correlation of the opticalamplitudes at frequencies ƒ_(c)±α₀/2 and a conventional spectralanalysis of the combined signal and noise power, as shown in the exampleof FIG. 1 for a 100 Gb/s PM-QPSK noisy signal 30 compared with anoiseless signal 31. Spectral components 32 and 33 may be used formeasuring the correlation coefficients at optical frequencies ƒ_(c)−α₀/2and ƒ_(c)+α₀/2. The spectral components 32 and 33 may be preferablyselected so that differences between each of optical frequencies of thespectral components 32 and 33 and the carrier frequency ƒ_(c) of themodulated optical signal in the selected wavelength channel aresubstantially of equal magnitude α₀/2, so that the frequency intervalmay be substantially equal to the symbol repetition frequency, or aninteger multiple of the symbol repetition frequency. More generally, anytwo predetermined optical frequencies in a selected one of the pluralityof wavelength channels may be used, provided that the opticalfrequencies are separated by a non-zero frequency interval.

It should be noted that the above described technique does not requireforeknowledge of the time-varying waveform or the particular modulationformat of the transmitted optical signal. Therefore, determination ofthe in-band OSNR may not require any calibration with a noiselesssignal, e.g., the noiseless signal 31. The only foreknowledge requiredfor this technique may be acknowledging that the noiseless signalexhibits a spectral amplitude correlation at frequencies ƒ_(c)±α₀/2 withŜ_(x) ^(α) ⁰ (ƒ_(c))=1. While it may be advantageous to haveforeknowledge of the symbol period T_(symbol), of the modulated signal,in order to determine the frequencies ƒ_(c)±α₀/2, this information maynot be required when Ŝ_(x) ^(α)(ƒ_(c)) is measured at a multitude offrequency pairs ƒ_(c)±α/2, with α ideally ranging from 0 to the largestpossible value, and when

P_(N)

is determined from the maximal value of Ŝ_(x) ^(α)(ƒ_(c)) observed inthis multitude of measurements. Advantageously, the frequency rangewhere Ŝ_(x) ^(α)(ƒ_(c)) is expected to be maximal may be determined froma simple analysis of the signal's power spectrum. Preferably, thefrequency interval between the measured spectral components, e.g., thespectral components 32 and 33 of FIG. 1 , may be substantially equal tothe symbol repetition frequency of the modulated optical signal in theselected wavelength channel, or an integer multiple thereof. It isfurther preferable that differences between each of the first and secondoptical frequencies of the measured spectral components, and the carrierfrequency ƒ_(c) of the modulated optical signal in the selectedwavelength channel may be substantially of equal magnitude.

The in-band OSNR measurement method as disclosed herein, for example,may be applied to transmitted signals that are encoded with chirp-freeASK, BPSK, ordinary QPSK, and other higher-order M-ary QAM formatswithout requiring detailed knowledge of the particular modulation formatencoded in the analyzed noisy signal. Referring to FIG. 2 , forinstance, a numerical simulation of the in-band OSNR determined from theabove equation is shown fora 100 Gb/s polarization-multiplexed QPSKsignal. It can be seen from FIG. 2 that an estimated in-band OSNR 40 issubstantially equal to the reference OSNR over a range from at least 0dB to 30 dB. A normalized SCDF 41 is plotted for a reference.

According to an example, the in-band OSNR of a transmitted noisy signalmay be determined from a measurement of the spectral correlation of theoptical amplitudes at any combination of frequencies ƒ±α/2 for whichŜ_(x) ^(α)(ƒ)=1. It should be appreciated that digitally modulatedsignals encoded with ASK, BPSK, QPSK, or 16-QAM, for example, mayexhibit Ŝ_(x) ^(α) ⁰ (ƒ)=1, as long as the two optical frequenciesƒ−α₀/2 and ƒ+α₀/2 are within the optical bandwidth of the modulatedsignal. This is illustrated in FIG. 3 for the example of a 100 Gb/s PMRZ-QPSK noisy signal 50, compared with a noiseless signal 51. In FIG. 3, optical frequencies 52 and 53, at which the time-varying signalmeasurements are performed, may be offset from the carrier frequencyƒ_(c). However, it should be appreciated that the bandwidth of thetransmitted optical signal may be limited by the spectral width of theDWDM channel. For example, when 50 Gb/s PM-BPSK, 100 Gb/s PM-QPSK, or200 Gb/s PM-16-QAM signals (all having T_(symbol)=40 ps) are transmittedthrough a 50-GHz wide DWDM channel, the useful values off may berestricted to the range ƒ_(c)−α₀/2<ƒ<ƒ_(c)+α₀/2.

In the case of ƒ≠ƒ_(c), as shown in FIG. 3 , it should be appreciatedthat

P_(S)(ƒ+α₀/2)

≠

P_(S)(ƒ−α₀/2)

, so that the process of determining the OSNR may become morecomplicated. For example, denoting

P_(S)(ƒ+α₀/2)

=C

P_(S)(ƒ−α₀/2)

≡C

P_(S)

, with C>0 being a real number, the normalized SCDF may be representedby

${{{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)} = \frac{\sqrt{C}\left\langle P_{S} \right\rangle}{\sqrt{\left\langle {{CP}_{S} + P_{N}} \right\rangle}\sqrt{\left\langle {P_{S} + P_{N}} \right\rangle}}},$which may be solved analytically or numerically for

P_(N)

/(

P_(S)

+

P_(N)

).

However, at large OSNR values, it should be appreciated that

P_(S)

>>

P_(N)

, so that the normalized SCDF may be approximated as

${{{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)} \approx \frac{\left\langle P_{S} \right\rangle^{2}}{\left\langle P_{S} \right\rangle^{2} + {\frac{C + 1}{2C}\left\langle P_{S} \right\rangle\left\langle P_{N} \right\rangle} + \frac{\left\langle P_{N} \right\rangle^{2}}{2C}}},$from which the average noise power

P_(N)

at the two frequencies ƒ±α₀/2 may be readily calculated as

$\begin{matrix}{\frac{\left\langle P_{N} \right\rangle}{\left\langle {P_{S}\left( {f - {\alpha_{0}/2}} \right)} \right\rangle} = {{- \frac{C + 1}{2}} + \sqrt{\left( \frac{C + 1}{2} \right)^{2} + {2C\frac{1 - {{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}{{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}}}} \\{{\approx {\frac{2C}{C + 1} \cdot \frac{1 - {{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}{{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}},}\end{matrix}$or more conveniently as

${\frac{\left\langle P_{N} \right\rangle}{\left\langle {{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle} \approx \frac{2{C\left\lbrack {1 - {{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}} \right\rbrack}}{{2C} + {\left( {1 - C} \right){{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}}},$so that the in-band OSNR may be represented as

$\begin{matrix}{{OSNR} \approx {{\frac{\sum\limits_{i}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle B_{meas}}}{\left\langle {{P_{S}\left( {f_{c} - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle B_{noise}} \cdot \frac{1 + {\frac{1 - C}{2C}{{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}}{1 - {{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}} - {\sum\limits_{i}\frac{B_{meas}}{B_{noise}}}}} \\{= \frac{\sum\limits_{i}{\begin{Bmatrix}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle - \left\langle {{P_{S}\left( {f_{c} - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle} \\\frac{2{C\left\lbrack {1 - {{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}} \right\rbrack}}{{2C} + {\left( {1 - C} \right){{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}}\end{Bmatrix}B_{meas}}}{\left\langle {{P_{S}\left( {f_{c} - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle\frac{2{C\left\lbrack {1 - {{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}} \right\rbrack}}{{2C} + {\left( {1 - C} \right){{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}}B_{noise}}}\end{matrix}$

Therefore, in-band OSNR of a transmitted noisy signal may be determinedfrom a measurement of the spectral correlation of the optical amplitudesat arbitrary frequencies ƒ±α₀/2 and an additional measurement of theoptical power spectrum of the noisy signal. Again, this technique,advantageously, may not require foreknowledge of the particularmodulation format, bit-rate, or time-varying waveform of the transmittedsignal. If the symbol period 1/α₀ of the signal is unknown, a suitablefixed value may be chosen for ƒ−α₀/2, and then Ŝ_(x) ^(α)(ƒ) may bemeasured at a multitude of frequency pairs {ƒ−α₀/2,ƒ+α/2}, with aranging from 0 to the largest expected value.

P_(N)

may then be determined from a maximal value of Ŝ_(x) ^(α)(ƒ_(c))observed in these measurements.

However, accurate determination of the in-band OSNR may requireforeknowledge of the ratio C of the two noiseless signal powers atƒ±α₀/2, which may be determined from a measurement of the optical powerspectrum of the noiseless signal (e.g. directly after the transmitter).If such measurement is not available, the first-order approximation maybe used

${{C \approx C_{1}} = \frac{\left\langle {{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle}{\left\langle {{P_{S}\left( {f + {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle}},$to estimate the in-band OSNR in the signal,

${OSNR} \approx {{\frac{\sum\limits_{i}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle B_{meas}}}{\left\langle {{P_{S}\left( {f_{c} - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle B_{noise}} \cdot \frac{1 + {\frac{1 - C_{1}}{2C_{1}}{{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}}{1 - {{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}} - {\sum\limits_{i}{\frac{B_{meas}}{B_{noise}}.}}}$

Referring to FIG. 4 , an example may be shown of OSNR curves 60A and 60Bestimated from a numerically simulated noisy 100 Gb/s PM-QPSK signalhaving an OSNR of 15 dB. The graph may also display a normalizedamplitude SCDF 61 as a function of the frequency offset ƒ−ƒ_(c). OSNR60A has been calculated from the above equation using the first-orderapproximation C≈C₁ (the SCDF 61 may be shown in bold and the OSNR 60Amay be shown by solid curves). It may be appreciated that thisapproximation may slightly overestimate the OSNR at offset frequenciesbeyond 8 GHz, where the signal power

P_(S)(ƒ+α₀/2)

may be small and comparable to the noise power

P_(N)

, as seen in FIG. 3 . For comparison, the dashed curve 60B in FIG. 4 maydisplay the in-band OSNR calculated from the exact formula for C=1. Itis evident from this curve that this formula may lead to a substantialoverestimation of the OSNR, e.g., by up to and more than 2 dB.

If the first-order approximation of C is not deemed to be accurateenough to determine the in-band OSNR, a second-order approximation maybe employed, which may be obtained by subtracting the first-orderaverage noise power

$\left\langle P_{N1} \right\rangle \approx {\left\langle {{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle\frac{2{C_{1}\left\lbrack {1 - {{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}} \right\rbrack}}{{2C_{1}} + {\left( {1 - C_{1}} \right){{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}}}$from the two noisy signal powers measured at frequencies ƒ±α₀/2 and byrecalculating C as

${{C \approx C_{2}} = \frac{\left\langle {{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle - \left\langle P_{N1} \right\rangle}{\left\langle {{P_{S}\left( {f + {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle - \left\langle P_{N1} \right\rangle}},$which may then be used for an improved second-order approximation of thein-band OSNR,

${OSNR} \approx {{\frac{\sum\limits_{i}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle B_{meas}}}{\left\langle {{P_{S}\left( {f_{c} - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle B_{noise}} \cdot \frac{1 + {\frac{1 - C_{2}}{2C_{2}}{{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}}{1 - {{\overset{\hat{}}{S}}_{x}^{\alpha_{0}}(f)}}} - {\sum\limits_{i}{\frac{B_{meas}}{B_{noise}}.}}}$The above described technique may be iterated multiple times, as needed,until a desired accuracy of the in-band OSNR is obtained.

Amplitudes of the spectral components of a transmitted signal may dependquite sensitively on signal distortions caused by chromatic dispersion(CD) or polarization-mode dispersion (PMD), which the modulated signalmay have experienced in the transmission link prior to being analyzed atthe OSNR monitoring point. Group velocity dispersion (GVD) from CD, forexample, may introduce a differential phase shift between the spectralamplitudes of the Jones vectors X_(T)(t,ƒ+α/2) and X_(T)(t,ƒ+α/2),whereas PMD-induced differential group delays (DGDs) may introduce adifferential polarization transformation between the two Jones vectors.Consequently, the correlation between the two spectral components maybecome severely distorted, so that Ŝ_(x) ^(α) ⁰ (ƒ)<1 even for noiselesssignals. Therefore, uncompensated GVD and/or DGD in the noisy signal maylead to a substantial underestimation of the in-band OSNR when using thespectral correlation method disclosed above.

Fortunately, the differential phase shifts caused CD and thedifferential polarization transformation caused by PMD may becompensated for by artificially introducing differential phase shiftsand/or differential polarization transformations in the measured Jonesvectors X_(T)(t,ƒ+α₀/2) and X_(T)(t,ƒ−α₀/2), and by varying these phaseshifts and/or polarization transformations until Ŝ_(x) ^(α) ⁰ (ƒ) ismaximal. To that end, determining the correlation may include (i)removing differential phase and time delays introduced by chromaticdispersion in the transmission link between the time-varying parametersat the optical frequencies, at which the measurement is performed; and(ii) removing a differential group delay introduced by polarization modedispersion in the transmission link between the time-varying parametersat the first and second optical frequencies. Even if the GVD- andDGD-induced distortions in the signal amplitudes are not perfectlycompensated, such procedure may substantially reduce errors ofdetermining the in-band OSNR from the measured spectral correlation.

In another example, end-to-end GVD and DGD in the transmission link maybe determined from an adaptive compensation of the GVD and DGD in themeasured signal amplitudes using the above described algorithm formaximizing Ŝ_(x) ^(α) ⁰ (ƒ).

The in-band OSNR may also be determined from the spectral correlationsof the optical signal intensities (or signal powers) at two differentfrequencies, i.e. from the correlations between the spectral powercomponents {tilde over (P)}(t, v)=|{tilde over (X)}_(T)(t, v)|². Itfollows from the above considerations that the spectral components ofthe signal power exhibit strong correlations whenever the spectralcomponents of the signal amplitude are strongly correlated.Consequently, modulated optical signals encoded with ASK, BPSK, ordinaryQPSK and 16-QAM formats exhibit strong correlations of the time-varyingoptical power components at frequencies ƒ±α₀/2. An example of the strongcorrelations between the time-varying signal powers 71, 72 at ƒ±α₀/2 maybe shown in FIG. 5 for a noiseless 100 Gb/s PM-QPSK signal. It may beseen from FIG. 5 that the power variations 71, 72 at the two frequenciesmay overlap completely, e.g., they are substantially identical.

It should be appreciated that signals encoded with ASK and BPSKmodulation may exhibit additional correlations of the spectral powercomponents beyond those found for the spectral amplitude components. Itmay be shown, for example, that the spectral power components of ASK andBPSK signals are correlated at any two pairs of frequencies ƒ_(c)±α/2within the bandwidth of the signal, e.g., for arbitrary offset frequencyα.

Just like for the optical amplitudes, spectral components of the opticalpower levels of random ASE noise may not exhibit any significantcorrelations. When optical noise is added to a modulated signal,therefore, the correlation between the spectral components of the noisysignal power may decrease with decreasing OSNR. An example of thereduced spectral correlations between the optical powers at ƒ_(c)±α₀/2may be shown in FIG. 6 for a noisy 100 Gb/s PM-QPSK signal having anOSNR of only 12 dB. It may be seen from this graph that variations 81,82 of the two optical powers with time may be substantially different.

Furthermore, it should be appreciated that the spectral correlations ofthe optical signal power may be much less sensitive to waveformdistortions caused by PMD and CD than the correlations of the opticalsignal amplitudes. In general, GVD from CD and/or DGD from PMD mayintroduce a differential time delay Δt between the time-varying opticalpower measurements {tilde over (P)}(t, ƒ−α/2) and {tilde over(P)}(t,ƒ+α/2), leading to signals of the form {tilde over(P)}(t−Δt/2,ƒ−α/2) and {tilde over (P)}(t+αt/2, ƒ+α/2), as shown in FIG.7 for the example of a 100 Gb/s PM-QPSK signal distorted by 50 ns/nmGVD. Two curves 91, 92 in FIG. 7 may be shifted in time by about 10 ns,but are otherwise substantially identical. Accordingly, thesedifferential time delays may substantially reduce the spectralcorrelations in a noise-free signal and, hence, may need to becompensated prior to calculating the correlation between the twosignals. This may be accomplished, for example, by introducing avariable time-delay between the two signals and by adaptively varyingthis delay until the correlation between the two signals is maximal.

In an example, it may be advantageous to define a normalized power SCDFfor the spectral power components {tilde over (P)}(t,ƒ+α/2)=|{tilde over(X)}_(T)(t,ƒ+α/2)|² and {tilde over (P)}(t,ƒ+α/2)=|{tilde over(X)}_(T)(t,ƒ+α/2)|² of the noisy signal, analogous to the normalizedamplitude SCDF described above:

${{\overset{\hat{}}{S}}_{p}^{\alpha}(f)} \equiv {\frac{\left\langle {{\overset{˜}{P}\left( {t,{f + {\alpha/2}}} \right)} \cdot {\overset{˜}{P}\left( {t,{f - {\alpha/2}}} \right)}} \right\rangle}{\sqrt{\left\langle \left\lbrack {\overset{˜}{P}\left( {t,{f + {\alpha/2}}} \right)} \right\rbrack^{2} \right\rangle}\sqrt{\left\langle \left\lbrack {\overset{˜}{P}\left( {t,{f - {\alpha/2}}} \right)} \right\rbrack^{2} \right\rangle}}.}$It may be shown that noiseless signals encoded with ASK-, BPSK-,ordinary QPSK- and 16-QAM modulation exhibit Ŝ_(p) ^(α) ⁰ (ƒ)=1 at anytwo frequency pairs ƒ±α₀/2 which may be within the optical bandwidth ofthe transmitted signal. For modulated signals with added ASE noise, thenormalized power SCDF may generally be less than unity, e.g., Ŝ_(p) ^(α)⁰ (ƒ)<1 just like the normalized amplitude SCDF. The normalized powerSCDF for noisy signals may be calculated analytically as describedbelow.

In the case of α>>1/T, which is of interest in the present disclosure,the numerator of Ŝ_(p) ^(α)(ƒ) may be expanded into the following terms

{tilde over (P)}(t,ƒ+α/2)·{tilde over (P)}(t,ƒ−α/2)

=

P _(S)(t,ƒ+α/2)·P _(S)(t,ƒ−α/2)

+

P _(S)(t,ƒ+α/2)·P _(N)(t,ƒ−α/2)

+

P _(N)(t,ƒ+α/2)·P _(S)(t,ƒ−α/2)

, +

P _(N)(t,ƒ+α/2)·P _(N)(t,ƒ−α/2)

wherein

P _(S)(t,ƒ+α/2)·P _(N)(t,ƒ−α/2)

=

P _(S)(t,ƒ+α/2)

·

P _(N)(t,ƒ−α/2)

,and likewise

P _(N)(t,ƒ+α/2)·P _(S)(t,ƒ−α/2)

=

P _(N)(t,ƒ+α/2)

·

P _(S)(t,ƒ−α/2)

,because the time-varying signal amplitudes may not be correlated withthe random variations of the noise amplitudes. Furthermore, since therandom noise power variations at substantially different opticalfrequencies may be uncorrelated, this may result in the followingexpression

P _(N)(t,ƒ+α/2)·P _(N)(t,ƒ−α/2)

=

P _(N)(t,ƒ+α/2)

·

P _(N)(t,ƒ−α2)

=

P _(N)(t,ƒ+α/2)

².Similarly, the two terms in the denominator of Ŝ_(p) ^(α)(ƒ) may beexpressed as

[{tilde over (P)}(t,ƒ+α/2)]²

=

[P _(S)(t,ƒ+α/2)]²

+

[P _(N)(t,ƒ+α/2)]²

+3

P _(S)(t,ƒ+α/2)

·

P _(N)(t,ƒ+α/2)

[{tilde over (P)}(t,ƒ−α/2)]²

=

[P _(S)(t,ƒ−α/2)]²

+

[P _(N)(t,ƒ−α/2)]²

+3

P _(S)(t,ƒ−α/2)

·

P _(N)(t,ƒ−α/2)

wherein the third term on the right side of these equations may includethe contribution

{Re[X _(T)(t,ƒ±α/2)·N _(T)*(t,ƒ±α/2)]}²

=¼

P _(S)(t,ƒ±α/2)

·

P _(N)(t,ƒ±+/2)

.

Therefore, in the special case of ƒ=ƒ_(c) and α=α₀, the power SCDF maybe written in the form

${{{\overset{\hat{}}{S}}_{p}^{\alpha_{0}}\left( f_{c} \right)} = \frac{\left\langle {P_{S}^{2}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle + {2{\left\langle {P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle \cdot \left\langle P_{N} \right\rangle}} + \left\langle P_{N} \right\rangle^{2}}{\left\langle {P_{S}^{2}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle + {3{\left\langle {P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle \cdot \left\langle P_{N} \right\rangle}} + \left\langle P_{N}^{2} \right\rangle}},{wherein}$⟨P_(N)⟩ = ⟨P_(N)(t, f_(c) + α₀/2)⟩ = ⟨P_(N)(t, f_(c) − α₀/2)⟩, ⟨P_(N)²⟩ = ⟨[P_(N)(t, f_(c) + α₀/2)]²⟩ = ⟨[P_(N)(t, f_(c) − α₀/2)]²⟩.The above expression for power SCDF may contain two unknown andpotentially large quantities,

P_(S) ²

and

P_(N) ²

, which may not be directly measured or obtained from the measurementsof the combined signal and noise powers at frequencies ƒ+α/2 and ƒ−α/2.

It should be appreciated that the averaged squared noise power

P_(N) ²

may related to the average noise power

P_(N)

as

P_(N) ²

=1.5

P_(N)

², because of the statistical properties of random ASE noise. In anexample, optical ASE noise may be described as a Gaussian randomprocess, which remains a Gaussian random process even after arbitrarylinear optical filtering (e.g., before or in the optical receiver).Thus, the relation

P_(N) ²

=1.5

P_(N)

² may be obtained from the known second and fourth moments of a Gaussianrandom process.

Furthermore, according to an example,

P_(S) ²

may be eliminated by multiplying the complementary power SCDF,

${{1 - {{\overset{\hat{}}{S}}_{p}^{\alpha_{0}}\left( f_{c} \right)}} = \frac{\left\langle P_{N}^{2} \right\rangle - \left\langle P_{N} \right\rangle^{2} + {\left\langle {P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle \cdot \left\langle P_{N} \right\rangle}}{\left\langle {P_{S}^{2}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle + {3{\left\langle {P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle \cdot \left\langle P_{N} \right\rangle}} + \left\langle P_{N}^{2} \right\rangle}},$with the dimensionless factor D(ƒ_(c),α₀), defined as

${{D\left( {f,\alpha} \right)} \equiv \frac{\sqrt{\left\langle \left\lbrack {\overset{˜}{P}\left( {t,{f + {\alpha/2}}} \right)} \right\rbrack^{2} \right\rangle}\sqrt{\left\langle \left\lbrack {\overset{˜}{P}\left( {t,{f - {\alpha/2}}} \right)} \right\rbrack^{2} \right\rangle}}{\left\langle {\overset{˜}{P}\left( {t,{f + {\alpha/2}}} \right)} \right\rangle \cdot \left\langle {\overset{˜}{P}\left( {t,{f - {\alpha/2}}} \right)} \right\rangle}},$which may be readily calculated from the measured signal and noisepowers {tilde over (P)}(t,ƒ±α/2).

At optical frequencies ƒ_(c)+α₀/2 and ƒ_(c)−α₀/2, this expression may beobtained

${{D\left( {f_{c},\alpha_{0}} \right)} = \frac{\rangle{P_{S}^{2}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\left\langle {+ 3} \right\rangle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\left\langle \cdot \right\rangle P_{N}\left\langle + \right\rangle P_{N}^{2}\langle}{\rangle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\left\langle {}^{2}{+ 2} \right\rangle{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)}\left\langle \cdot \right\rangle P_{N}\left\langle + \right\rangle P_{N}\left\langle {}^{2} \right.}},$and after multiplication with the complementary power SCDF, it may beexpressed as follows

${{{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)} \equiv {\left\lbrack {1 - {{\overset{\hat{}}{S}}_{p}^{\alpha_{0}}\left( f_{c} \right)}} \right\rbrack{D\left( {f_{c},\alpha_{0}} \right)}}} = \frac{{{0.5}\left\langle P_{N} \right\rangle^{2}} + {\left\langle {P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} \right\rangle \cdot \left\langle P_{N} \right\rangle}}{\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle^{2}}}.$This equation may then be solved for

P_(N)

${\frac{\left\langle P_{N} \right\rangle}{\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle} = {1 - \sqrt{1 - {2{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}}}}},$and substituted into the equation for OSNR to produce

$\begin{matrix}{{OSNR} = {{\frac{\sum\limits_{i}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle B_{meas}}}{\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle B_{noise}} \cdot \frac{1}{1 - \sqrt{1 - {2{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}}}}} -}} \\{\sum\limits_{i}\frac{B_{meas}}{B_{noise}}} \\{{= \frac{\sum\limits_{i}{\begin{Bmatrix}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle - \left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle} \\\left\lbrack {1 - \sqrt{1 - {2{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}}}} \right\rbrack\end{Bmatrix}B_{meas}}}{{\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle\left\lbrack {1 - \sqrt{1 - {2{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}}}} \right\rbrack}B_{noise}}},}\end{matrix}$wherein again the summation may extend over the entire spectral width ofthe transmitted optical signal.

Therefore, the in-band OSNR in a noisy signal may be determined from aspectral correlation measurement of the signal powers at opticalfrequencies ƒ_(c)+α₀/2 and ƒ_(c)−α₀/2. Just like in the case of usingcorrelation measurements of the signal's amplitude, these measurementsmay not require foreknowledge of the signal's modulation format,bitrate, or waveform, which is advantageous for the reasons describedherein.

For large OSNR values (e.g., larger than 15 dB for a 100 Gb/s PM RZ-QPSKsignal), one may approximate the in-band OSNR by the simpler expression

$\begin{matrix}{{OSNR} \approx {{\frac{\sum\limits_{i}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle B_{meas}}}{\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle B_{noise}} \cdot \left\lbrack {\frac{1}{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)} - 1} \right\rbrack} - {\sum\limits_{i}\frac{B_{meas}}{B_{noise}}}}} \\{\approx {{\frac{\sum\limits_{i}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle B_{meas}}}{\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle B_{noise}} \cdot \frac{1}{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}} - {\sum\limits_{i}\frac{B_{meas}}{B_{noise}}}}} \\{{= \frac{\sum\limits_{i}{\left\{ {\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle - {\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}}} \right\}\mspace{11mu} B_{meas}}}{\left\langle {{P_{S}\left( {f_{c} \pm {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}\left( f_{c} \right)}B_{noise}}},}\end{matrix}$because

P_(N)

<<

P_(S)

, so that Ŝ_(p) ^(α) ⁰ (ƒ_(c))≈1 and hence {circumflex over (D)}_(p)^(α) ⁰ (ƒ_(c))<<1. FIG. 8 displays a numerical simulation of thereference in-band OSNR 100A determined from the above equation fora 100Gb/s PM-QPSK signal. The reference in-band OSNR 100A is shown with athick solid line. A normalized power SCDF 101 at ƒ_(c)±α₀/2 may also beshown. It should be appreciated that an estimated in-band OSNR 100B,shown with a dashed line, may be substantially equal to the referencein-band OSNR 100A over a range from about 15 dB to at least 30 dB.

Importantly, in-band OSNR may be determined from a spectral correlationmeasurement of transmitted signal powers at any frequency pair ƒ±α₀/2within the spectral bandwidth of the signal, similar to the spectralcorrelation measurement of the transmitted signal amplitudes describedabove. Letting again

P_(S)(ƒ+α₀/2)

=C

P_(S)(ƒ−α₀/2)

≡C

P_(S)

, with C>1 being a real number, the normalized SCDF of the signal powersmay take the general form

${{{\overset{\hat{}}{S}}_{p}^{\alpha_{0}}(f)} = \frac{{C\left\langle P_{S}^{2} \right\rangle} + {\left( {1 + C} \right){\left\langle P_{S} \right\rangle \cdot \left\langle P_{N} \right\rangle}} + \left\langle P_{N} \right\rangle^{2}}{\sqrt{\left\langle P_{S}^{2} \right\rangle + {3{\left\langle P_{S} \right\rangle \cdot \left\langle P_{N} \right\rangle}} + \left\langle P_{N}^{2} \right\rangle}\sqrt{{C^{2}\left\langle P_{S}^{2} \right\rangle} + {3C{\left\langle P_{S} \right\rangle \cdot \left\langle P_{N} \right\rangle}} + \left\langle P_{N}^{2} \right\rangle}}},$

wherein

P_(N) ²

=1.5

P_(N)

². It may be possible to eliminate the unknown quantity

P_(S) ²

from the above equation, using the dimensionless factor

${{D\left( {f,\alpha_{0}} \right)} = \frac{\sqrt{\left\langle P_{S}^{2} \right\rangle + {3{\left\langle P_{S} \right\rangle \cdot \left\langle P_{N} \right\rangle}} + \left\langle P_{N}^{2} \right\rangle}\sqrt{{C^{2}\left\langle P_{S}^{2} \right\rangle} + {3C{\left\langle P_{S} \right\rangle \cdot \left\langle P_{N} \right\rangle}} + \left\langle P_{N}^{2} \right\rangle}}{\left\langle {{CP_{S}} + P_{N}} \right\rangle\left\langle {P_{S} + P_{N}} \right\rangle}},$but the results tend to be fairly complex.

For signals with relatively large OSNR (e.g., above 15 dB for a 100 Gb/sPM RZ-QPSK signal), the terms proportional to

P_(N)

² and

P_(N)

² may be neglected or treated as negligible, so that the complementarySCDF may be simplified to

${{1 - {{\overset{\hat{}}{S}}_{p}^{\alpha_{0}}(f)}} \approx \frac{0.5\left( {1 + C} \right){\left\langle P_{S} \right\rangle \cdot \left\langle P_{N} \right\rangle}}{\sqrt{\left\langle P_{S}^{2} \right\rangle + {3{\left\langle P_{S} \right\rangle \cdot \left\langle P_{N} \right\rangle}}}\sqrt{{C^{2}\left\langle P_{S}^{2} \right\rangle} + {3C{\left\langle P_{S} \right\rangle \cdot \left\langle P_{N} \right\rangle}}}}},$so that

${{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}(f)} \equiv {\left\lbrack {1 - {{\overset{\hat{}}{S}}_{p}^{\alpha_{0}}(f)}} \right\rbrack{D\left( {f,\alpha_{0}} \right)}} \approx {\frac{0.5\left( {1 + C} \right){\left\langle P_{S} \right\rangle \cdot \left\langle P_{N} \right\rangle}}{{C\left\langle P_{S} \right\rangle^{2}} + {\left( {C + 1} \right){\left\langle P_{S} \right\rangle \cdot \left\langle P_{N} \right\rangle}}}.}$This expression may be readily solved for

P_(N)

, yielding

${\left\langle P_{N} \right\rangle \approx {\frac{2C{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}(f)}}{1 + C - {2{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}(f)}}}\left\langle {P_{S} + P_{N}} \right\rangle}},$and subsequently substituted into the equation for the OSNR:

$\begin{matrix}{{OSNR} \approx {{\frac{\sum\limits_{i}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle B_{meas}}}{\left\langle {{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle B_{noise}}.\left\{ {\frac{1 + C}{2C{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}(f)}} - \frac{1}{C}} \right\}} - {\sum\limits_{i}\frac{B_{meas}}{B_{noise}}}}} \\{\approx {{\frac{\sum\limits_{i}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle B_{meas}}}{\left\langle {{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle B_{noise}} \cdot \frac{1 + C}{2C{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}(f)}}} - {\sum\limits_{i}\frac{B_{meas}}{B_{noise}}}}} \\{= {\frac{\sum\limits_{i}{\begin{Bmatrix}{\left\langle {{P_{S}\left( f_{i} \right)} + P_{N}} \right\rangle - \frac{2C{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}(f)}}{1 + C - {2{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}(f)}}}} \\\left\langle {{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle\end{Bmatrix}B_{meas}}}{\frac{2C{{\overset{\hat{}}{D}}_{p}^{\alpha_{0}}(f)}}{1 + C}\left\langle {{P_{S}\left( {f - {\alpha_{0}/2}} \right)} + P_{N}} \right\rangle B_{noise}}.}}\end{matrix}$

FIG. 9 shows an example of OSNR curves 110A, 110B estimated from anumerically simulated noisy 100 Gb/s PM-QPSK signal having an OSNR of 15dB. The graph also displays a normalized power SCDF 111 as a function ofthe frequency offset ƒ−θ_(c). The OSNR 110A has been calculated from theabove equation, where C may be calculated directly from the noisy signalspectrum, e.g., C≈

P_(S) (ƒ−α₀/2)+P_(N)

/

P_(S)(ƒ+α₀/2)+P_(N)

(the OSNR 110A may be shown with a bold solid line). It can be seen fromthis graph that the above approximation slightly underestimates the OSNRat offset frequencies beyond 8 GHz, where the magnitude of the signalpower

P_(S)(ƒ+α₀/2)

becomes comparable to that of the noise power

P_(N)

, as shown in FIG. 3 , so that terms proportional to

P_(N)

² may not be neglected and the above approximation for the in-band OSNRmay no longer be valid. For comparison, the dashed curve 110B in FIG. 9may display the in-band OSNR calculated from the formula derived forC=1.

Amplitudes and phases of the spectral signal components, preferably atoptical frequencies ƒ+α₀/2 and ƒ−α₀/2 as described above, may bedetected simultaneously by two parallel coherent optical receivers, eachhaving phase and polarization diversity. With reference to FIG. 10 , anexemplary apparatus 100 for determining OSNR of a modulated opticalsignal 121 may generally include a frequency selective splitter 122, ameasuring unit 123, and a digital signal processor (DSP) 124. Themodulated optical signal 121 contains a plurality of wavelengthchannels, e.g. wavelength division multiplexed (WDM) channels, or denseWDM (DWDM) channels, and is normally obtained at a test point along thecommunication link. An optional spectrum analyzer, not shown, may beused for measuring an optical power spectrum of the modulated opticalsignal 121. To be useful in the measurement, the optical power spectrummay have at least one of the plurality of wavelength channels.

In an example, the frequency selective splitter 122 may a dual-channelcoherent receiver having a tunable local oscillator light source, e.g.,a pair of cw tunable lasers (“local oscillators”, or LO lasers) 125A and125B tuned to first and second frequencies, for example ƒ+α₀/2 andƒ−α₀/2 respectively. The frequency selective splitter 122 may furtherinclude a 3 dB splitter 119A and four polarization beamsplitters (PBS)119B optically coupled, as shown. The modulated optical signal 121 maybe mixed with the corresponding local oscillator laser signals in hybridmixers 126. The measuring unit 123 may include differentialphotodetectors 127, low-pass filters (LPFs) 128, and analog-to-digitalconverters (ADC) 129. Other suitable configurations may be used.

In operation, the frequency selective splitter 122 may select portionsof the modulated optical signal 121 at first and second predeterminedoptical frequencies, in a selected one of the plurality of wavelengthchannels. The first and second predetermined optical frequencies may beseparated by a non-zero frequency interval, e.g. α₀ as explained above.To that end, the modulated optical signal 121 may be divided into twoidentical copies by means of the 3 dB power splitter 119A, which maythen be coupled into two substantially identical coherent receiver“channels”, e.g., receivers 122A and 122B. Each copy of the modulatedsignal 121 may be mixed with a highly coherent light from one of thetunable lasers 125A and 125B, outputting a highly coherentcontinuous-wave optical signal at a predetermined optical frequency. Thefrequencies of the two local oscillator lasers may be set to besubstantially equal to the two frequencies at which the spectralcorrelation is to be measured, e.g., to ƒ₁=ƒ+α₀/2 and ƒ₂=ƒ−α₀/2,respectively. The tunable lasers 125A and 125B may also have a narrowlinewidth, e.g., no greater than 100 kHz.

Prior to mixing with the output light of the tunable lasers 125A and125B, each of the two copies of the modulated signal may be decomposedinto two signals having mutually orthogonal polarization states by meansof the PBS 119B. Furthermore, each of the two orthogonally polarizedsignals may be further split into two identical copies and thenindependently mixed with the output light from the tunable LO lasers125A and 125B, by means of the 90° hybrid mixers 126. The first copy maybe mixed with the LO light having an arbitrary optical phase and thesecond copy with the LO light having an optical phase shifted by 90°relative to the LO light used for the first copy. Such a receiver may bereferred to herein as a coherent receiver having polarization and phasediversity.

The measuring unit 123 may measure time-varying optical amplitudes andphases of the first and second portions of the modulated optical signal121. To that end, the four different mixing products in each of the twocoherent receiver channels 122A and 122B may then be detected bybalanced photo-detectors 127, which convert the coherently mixed opticalamplitudes of the received signal and LO laser into a proportionalelectrical current, but substantially reject all non-coherently detectedsignal powers. The four time-varying detector currents i_(k)(t), k=1, .. . , 4, may be subsequently amplified and filtered by the fouridentical electrical LPFs 128, before they are converted into digitalsignals by a set of four high-resolution ADCs 129, and having aneffective number of bits (ENOB) of at least 12 and a sampling ratesubstantially higher than two times the bandwidth of LPFs.

Once the time-varying amplitudes and phases have been measured, thesignal processor 124 may determine a correlation between thetime-varying amplitudes and phases of the first and second portions ofthe modulated optical signal, and may calculate the OSNR from thecorrelation of the time-varying parameters and the power spectrum of themodulated optical signal 121. In an example, this may be done asfollows. The four receiver currents i₁(t,ƒ±α/2), . . . , i₄(t,ƒ±α/2),generated in each of the two coherent receiver channels 122A and 122B,may describe the amplitude, phase, and polarization state of thereceived optical signal at frequencies ƒ±α/2. The four currents may beused to form a two-dimensional complex Jones vector that is proportionalto the Jones vector of the optical signal, e.g.,

${{{\overset{˜}{X}}_{T}\left( {t,{f \pm {\alpha/2}}} \right)} = {\beta\left\lfloor \begin{matrix}{{i_{1}\left( {t,{f \pm {\alpha/2}}} \right)} + {{ji}_{2}\left( {t,{f \pm {\alpha/2}}} \right)}} \\{{i_{3}\left( {t,{f \pm {\alpha/2}}} \right)} + {{ji}_{4}\left( {t,{f \pm {\alpha/2}}} \right)}}\end{matrix} \right\rfloor}},$where β may be an undetermined proportionality factor and j=√{squareroot over (−1)}. Hence, the Jones vectors formed by the four receivercurrents in each of the two coherent receivers may be used to calculatethe normalized amplitude SCDF. In an example, this may be accomplishedin a fast digital signal processor (DSP), which processes the data atthe sampling rate of the ADC 129.

Prior to calculating the amplitude SCDF, any undesired differentialphase shifts between the two Jones vectors {tilde over (X)}_(T)(t,ƒ−α/2)and {tilde over (X)}_(T)(t,ƒ+α/2) 124C may be removed (or compensated)by digital signal processing, as described above, including differentialphase shifts introduced by CD in the communication systems (“CDcompensation”) as well as static phase shifts introduced in thereceiver. This may be achieved by a CD compensation module 124A.Likewise, any PMD-induced differential polarization transformationsbetween the two Jones vectors may be removed (“PMD compensation”). Thismay be achieved by a PMD compensation module 124B. This digital CD andPMD compensation may be accomplished iteratively by means of a feedbackloop, in which the differential phase shifts and the differentialpolarization transformations are varied in sufficiently small stepsuntil the amplitude SCDF reaches a maximum. Then, the normalizedamplitude correlation coefficient may be computed by a computing module124D.

For in-band OSNR measurements in QPSK and higher-order M-ary QAMsignals, the electrical bandwidth of the coherent receiver 122, whichmay be substantially equal to twice the bandwidth of the LPF 128, may beas small as possible, so that the correlation between the two detectedJones vectors is maximized. On the other hand, the receiver bandwidthmay be wide enough so that a sufficiently large electrical signal isavailable for determining the SCDF. In an example, the 3-dB electricalbandwidth of the LPF 128 may be around 100 MHz for in-band OSNRmeasurements in 40 Gb/s and 100 Gb/s PM-QPSK signals. FIG. 11 mayillustrate effects of the receiver bandwidth on the accuracy of OSNRmeasurements on a coherently detected 100 Gb/s PM RZ-QPSK signal for the3 dB bandwidth of the LPF 128 of 0.1 GHz, 0.4 GHz, 1 GHz, and 4 GHz.Severe degradations of the measurement accuracy may be seen at high OSNRvalues when the receiver bandwidth substantially exceeds 400 MHz. Itshould be appreciated that the bandwidth requirement for the LPF 128 mayscale linearly with the bit-rate of the signal 121.

Furthermore, the optical frequencies of the two tunable (LO) lasers 125Aand 125B may be precisely adjusted in order to measure maximalcorrelation between the spectral components of the analyzed opticalsignal 121. This may be accomplished by first setting one of the two LOlasers (e.g., 125A) to a fixed frequency within the spectral bandwidthof the signal, and by then varying the optical frequency of the other LOlaser (e.g., 125B) continuously—or in sufficiently small steps—until theSCDF reaches a maximum. It should be appreciated that a maximalamplitude correlation may be observed over a very narrow opticalfrequency range, e.g., of less than 1 MHz. FIG. 12 displays a numericalsimulation of an amplitude SCDF 141 as a function of the frequencyseparation of the two LO lasers 125A, 125B (FIG. 10 ) for the case of a100 Gb/s PM-QPSK signal. In contrast, maximal power correlation curves142A, 142B (FIG. 14 ) may be observed over a frequency range of severalMHz, which scales linearly with the electrical bandwidth of thereceiver. Consequently, spectral correlation measurements may require LOlasers with high frequency stability and low frequency noise. Typically,LO lasers 125A, 125B having a linewidth of less than 10 kHz may be usedfor amplitude correlation measurements, whereas LO lasers with alinewidth of less than 100 kHz may be sufficient for power correlationmeasurements. Solid curves 141, 142A correspond to an electricalreceiver with fifth-order Butterworth low-pass filter, and the dashedcurve 142B may correspond to a receiver with fifth-order Bessel low-passfilter, both having a 3-dB bandwidth of 40 MHz.

The embodiment of FIG. 10 may also be used to measure a power spectrumof the noisy signal (c.f. FIG. 1 ). This measurement may be illustratedin FIG. 13 . In an apparatus 110, one of the two coherent receiverchannels 122A, 122B, for example, may be needed e.g. 122A, includingelements similar to those described above with reference to FIG. 10 .The desired power spectrum may be obtained by scanning the LO laser 125Aover the entire bandwidth of the signal 121, while recording the innerproduct of the Jones vector, i.e. |X_(T) (t, ƒ)|², as shownschematically in FIG. 13 . Time-varying optical power levels of thefirst and second portions of the modulated optical signal 121 may beused instead of the time-varying optical amplitudes and phases.

FIG. 14 displays a schematic diagram of an apparatus 100 for the abovedescribed in-band OSNR measurements using the power SCDF of the opticalsignal. Similar to the apparatus 100 of FIG. 10 , the apparatus 140 ofFIG. 14 may have two parallel coherent receiver channels 122A, 122B,which may be connected to a fast DSP 164. One difference may be that theDSP in FIG. 14 may use modules 164C, 164D to calculate the correlationbetween the time-varying powers of the two Jones vectors 164A, e.g.,

$\begin{matrix}{{\overset{˜}{P}\left( {t,{f \pm {\alpha/2}}} \right)} = \left| {{\overset{˜}{X}}_{T}\left( {t,{f \pm {\alpha/2}}} \right)} \right|^{2}} \\{= {\beta^{2}\left\lbrack \left| {i_{1}\left( {t,{f \pm {\alpha/2}}} \right)} \middle| {}_{2}{+ \left| {i_{2}\left( {t,{f \pm {\alpha/2}}} \right)} \middle| {}_{2} + \right.} \right. \right.}} \\{\left. \left| {i_{3}\left( {t,{f \pm {\alpha/2}}} \right)} \middle| {}_{2}{+ \left| {i_{4}\left( {t,{f \pm {\alpha/2}}} \right)} \right|^{2}} \right. \right\rbrack.}\end{matrix}$The requirements for the electrical bandwidth of the receiver may besimilar or identical to those described above for measurements of theamplitude SCDF of FIG. 11 .

Prior to calculating the power SCDF, any undesired differential timedelays between the two signal powers {tilde over (P)}(t, ƒ−α/2) and{tilde over (P)}(t, ƒ+α/2) may need to be removed by digital signalprocessing, as described above, including delays introduced by CD andPMD as well as those introduced in the receiver. This differential timedelay compensation may be accomplished iteratively by means of afeedback loop, in which the delay is varied in sufficiently small stepsuntil the power SCDF reaches a maximum. The differential time delaycompensation, for example, may be performed by a module 164B.

Furthermore, the DSP 164 may also provide Jones and/or Stokescomputations 164E between 164B and 164C. For example, it should beappreciated that mixer block with LO frequency ƒ−α/2 accordingly mayhave 4 ADC outputs which may be grouped as vector:α=(α₁,α₂,α₃,α₄)

Retarder A may be controllable by parameter α:

$A = \begin{pmatrix}{\cos\;\alpha} & {{- \sin}\;\alpha} & 0 & 0 \\{\sin\;\alpha} & {\cos\;\alpha} & 0 & 0 \\0 & 0 & {\cos\;\alpha} & {\sin\;\alpha} \\0 & 0 & {{- \sin}\;\alpha} & {\cos\;\alpha}\end{pmatrix}$

Rotator B may be controllable by parameter β:

$B = \begin{pmatrix}{\cos\;\beta} & 0 & {\sin\;\beta} & 0 \\0 & {\cos\;\beta} & 0 & {\sin\;\beta} \\{{- \sin}\;\beta} & 0 & {\cos\;\beta} & 0 \\0 & {{- \sin}\;\beta} & 0 & {\cos\;\beta}\end{pmatrix}$

Equalizer C may be controllable by parameter c (c being not too far from0):

$C = \begin{pmatrix}{1 + c} & 0 & 0 & 0 \\0 & {1 + c} & 0 & 0 \\0 & 0 & {1 - c} & 0 \\0 & 0 & 0 & {1 - c}\end{pmatrix}$

In this example, the ADC values α may be multiplied in ADC sample speedinto PDL compensated values α′=(α′₁, α′₂, α′₃, α′₄) by equation:α′=C′(c)·B(β)·A(α)·α

Where the parameters α, β, c may be moved slowly by Stokes parameters.

It should be appreciated that Stokes computation may as follows. Controlvalues may be derived from the Stokes Vector S=(S₀, S₁, S₂, S₃), whichmay be computed from the ADC sample products (e.g., short time)averaged, for averaging some hundred sample products might beappropriate:

$\left. \begin{pmatrix} \cdot \\c \\\beta \\\alpha\end{pmatrix}\leftarrow S \right. = {\begin{pmatrix}S_{0} \\S_{1} \\S_{2} \\S_{3}\end{pmatrix} = \begin{pmatrix}{\overset{\_}{a_{1}^{\prime\; 2}} + \overset{\_}{a_{2}^{\prime\; 2}} + \overset{\_}{a_{3}^{\prime\; 2}} + \overset{\_}{a_{4}^{\prime\; 2}}} \\{\overset{\_}{a_{1}^{\prime\; 2}} + \overset{\_}{a_{2}^{\prime\; 2}} - \overset{\_}{a_{3}^{\prime\; 2}} - \overset{\_}{a_{4}^{\prime\; 2}}} \\{{2\overset{\_}{a_{1}^{\prime}a_{3}^{\prime}}} + {2\overset{\_}{a_{2}^{\prime}a_{4}^{\prime}}}} \\{{2\overset{\_}{a_{1}^{\prime}a_{3}^{\prime}}} - {2\overset{\_}{a_{2}^{\prime}a_{4}^{\prime}}}}\end{pmatrix}}$

Therefore, to reach a stable ball transformation may be provided with afeedback loop. In other words, for a given PDL distortion, aproportional/integral control loop may be set up which principally maytarget to reach (S₁, S₂, S₃)→(0, 0, 0):c=c+S ₁ /S ₀β=β+S ₂ /S ₀α=α+S ₃ /S ₀It should be appreciated that Jones and/or Stokes computations 164E maybe implemented independently, as described herein, any number of times.For example, Jones and/or Stokes computations 164E may be implementedindependently for each of the two LO Lasers 125A and 125B. Other variousexamples may also be provided.

Turning now to FIG. 15 , an apparatus 110 may be used for measuringspectral correlations in power level or amplitude of the signal 121.Similar to apparatus 100 of FIG. 10 and 100 of FIG. 14 , the apparatus110 of FIG. 15 may include a frequency selective splitter 172 includingtwo parallel coherent receivers, or receiver channels 172A, 172B, and ameasuring unit 173 to measure correlations between two distinctfrequency components of the optical amplitude or power of the signal121. Unlike the homodyne receivers used in the apparatus 100 of FIG. 10and 100 of FIG. 14 , the apparatus 110 of FIG. 15 may use two heterodynereceivers, in which the frequencies of the two LO lasers 125A and 125Bare offset from ƒ−α/2 and ƒ+α/2 by an equal amount ƒ_(IF), which issubstantially larger than the bandwidth of the signals from which theamplitude or power SCDF is calculated. As a result, the spectralcomponents to be analyzed may be found at intermediate frequency ƒ_(IF),in the receiver photo currents, which may have been generated by thecoherent mixer and, hence, can be selected by means of electricalband-pass filters (BPF) 178 having a pass band centered around ƒ_(IF).Advantageously, such heterodyne receivers do not require coherent mixerswith phase diversity, e.g., they do not require 90° hybrid mixers, butinstead may use simple 3 dB optical couplers 179A to superimpose theoptical signal with the output light of the LO lasers 125A, 125B,yielding two photo currents in each of the two coherent receivers 172A,172B.

The four signals needed to form the Jones vectors {tilde over(X)}_(T)(t,ƒ±α/2) in each of the two coherent receivers 172A, 172B maybe recovered by down-converting the two bandpass-filtered photocurrentsby means of an electrical homodyne receiver to the baseband.Advantageously, this down-conversion and subsequent electrical filteringmay be accomplished by digital signal processing in a DSP 174, as shownschematically in FIG. 15 , using digital signal mixers 175 and digitallow-pass filters 176. However, such digital down-conversion may requirethat the sampling rate of the ADCs 129 is substantially larger than 2ƒ_(IF).

In another example, the two independent LO lasers 125A, 125B for the tworeceivers 172A, 172B may be replaced by a single laser and an opticalfrequency shifter. Referring now to FIG. 16 , an apparatus 100 may besimilar to the apparatus 100 of FIG. 14 . In the apparatus 100 of FIG.16 , output light 181 of the LO laser 125A may be split (e.g., equally)into two signals 181A and 181B, of which the first signal 181A serves asthe LO signal for the first coherent receiver channel 122A, whereas thesecond signal 181B is shifted in frequency by an amount substantiallyequal to α by means of an optical frequency shifter 183, to generate aLO optical signal for the second coherent receiver channel 122B.

An advantage of using the single LO laser 125A for both coherent mixerpairs 126 may be that it substantially reduces the frequency stabilityrequirements for the LO laser 125A, provided that optical path lengthsfor the modulated signal between the 3 dB splitter 119A and the fourinputs to the 90° hybrid mixers 126 are substantially equal and,likewise, that the optical path lengths for the two LO signals betweenthe 3 dB splitter 179B and the four inputs to the 90° hybrid mixers 126are substantially equal. Another advantage of this embodiment may bethat the frequency difference between the two LO signals is determinedby the frequency offset generated in the optical frequency shifter 183,which may be set with much higher accuracy than the difference frequencyof two independent LO lasers 125A and 125B in FIGS. 10 and 14 .

Referring to FIG. 17A, an electro-optic frequency shifter 170A has beendescribed in detail by M. Izutsu et al. in “Integrated optical SSBmodulator/frequency shifter,” J. Quantum Electron. Vol. 17, p. 2225(1981), which is incorporated herein by reference, and will only bebriefly described herein. The electro-optic frequency shifter 170A mayinclude an optical modulator 193A coupled to a tunable laser source 194.The frequency offset, or the modulation frequency α may be determined bythe electrical frequency of a sinusoidal oscillator 191 outputting twosignals of identical frequency that are phase-shifted by 90°. Theoptical output signal of the frequency shifter 170A may be filtered byan optical band-pass filter 192, as shown in FIG. 17A, to filter outother undesired frequency components, e.g., harmonics of modulationfrequency α.

In FIGS. 17B and 17C, electro-optic frequency shifters 170B and 170C mayboth generate two shifted optical frequency components from a singleoptical input signal provided by the tunable laser 194. The frequencyshifter 170B of FIG. 17B may include a Mach-Zehnder modulator 193Bdriven by the sinusoidal oscillator 191 with a sinusoidal signal offrequency α/2, and operated in such a way that it generates acarrier-suppressed optical line spectrum, as disclosed by F. Heismann etal. in U.S. Pat. No. 8,135,275 issued Mar. 13, 2012 “Measuring chromaticdispersion in an optical wavelength channel of an optical fiber link”,incorporated herein by reference. The two main frequency components inthe output of the Mach-Zehnder modulator at frequencies ƒ−α/2 and ƒ+α/2may be separated from each other and the other undesired frequencycomponents by an optical diplexer 195.

The frequency shifter 170C shown in FIG. 17C may have two identicaloptical modulators 193A of the kind shown in FIG. 17A and, in general,generates fewer undesired frequency components that the frequencyshifter shown in FIG. 17B. However, it may have substantially higheroptical insertion loss than the simpler frequency shifter 170B of FIG.17B.

Turning now to FIG. 18 , another embodiment of an apparatus 180 formeasuring correlations in the signal's optical power spectrum is shown.The apparatus 180 includes two narrowband optical band-pass filters, 201(OBPF1) and 202 (OBPF2), for selecting spectral components to beanalyzed, two photodetectors 127, two LPFs 128, and two ADCs 129. Inoperation, optical power levels of the two spectral components filteredby the OBPF1 201 and OBPF2 202, may be detected by the photodetectors127, filtered by the LPFs 128, and digitized by the ADCs 129. Aprocessing unit 204 may perform differential group delay compensation(module 204A) and then calculates a normalized power correlationcoefficient (module 204B).

The apparatus 180 may be substantially simpler than the apparatus 140 ofFIG. 14 , but the apparatus 180 may require OBPFs with a 3-dB opticalbandwidth of less than 1 GHz. Furthermore, the OBPF1 201 and OBPF2 202may not exhibit significant polarization dependence, such aspolarization-dependent loss or polarization-dependent shifts of theoptical passband. The techniques described herein do not requireforeknowledge of the modulation format and bit-rate of the opticalsignal, nor is careful calibration of the apparatus with a noiselesssignal needed. In addition, techniques described herein may also providea reliable maximum correlation when polarization-dependent loss (PDL) ispresent on the fiber link.

Examples to mitigate polarization-dependent loss (PDL) when determiningin-band OSNR in signals substantially distorted by chromatic dispersion(CD) or polarization-mode dispersion (PMD) are disclosed herein. Ingeneral, because two optical frequencies for noise distributions mayconsist of two independent noise distributions, originating from the twopolarization multiplexed data streams, there may be strong polarizationrotation along the fiber. This may result in PDL that is unequal for thetwo optical frequencies, which in turn may result in unequaldistribution in the two optical frequencies. Ultimately, this may causesevere degradation in an OSNR measurement.

Typically, polarization state and the PDL may randomly vary in a 10 Hzcycle region. In order to mitigate PDL in such a situation, it may behelpful to retransform any elliptical polarization, introduced by thetransmission line, back to a ball polarization, e.g., in roughly 1 kHzspeed, and then subsequently correlating the two optical frequencies. Inthis case, an integration duration for determining the state ofpolarization (SOP) may be the correlation time interval, though notnecessarily. For each of the two optical frequencies, a transformer maybe used. A transformer may consist of Jones matrix multiplications, orother similar calculations, working as retarder, rotator, and/orequalizer. The output of the transformer may feed the correlator. Tocontrol the transformer, a Stokes vector of the output may becalculated.

In an example, the Stokes vector or parameter may include the following:S0 sum of power, S1 vertical/horizontal power unbalance, S2 45/−45degree power unbalance, S3 right/left circular power unbalance. S2, S3may be signed, and may be suitable to control the retarder, rotator,and/or equalizer. A feedback loop may also be setup to get S2, S3 tozero. It should be appreciated, in some examples, that S1 may controlthe equalizer, S2 may control the rotator, and S3 may controls theequalizer. In some examples, the Jones matrix computations may beachieved in ADC-sample speed, and the Stokes and feedback loopcomputations may be achieved for every integration interval. It shouldbe appreciated that these calculations and determinations may be carriedout in an FPGA, or other similar hardware or element.

The hardware used to implement the various illustrative logics, logicalblocks, modules, and circuits described in connection with the aspectsdisclosed herein may be implemented or performed with a general purposeprocessor, a digital signal processor (DSP), an application specificintegrated circuit (ASIC), a field programmable gate array (FPGA) orother programmable logic device, discrete gate or transistor logic,discrete hardware components, or any combination thereof designed toperform the functions described herein. A general-purpose processor maybe a microprocessor, but, in the alternative, the processor may be anyconventional processor, controller, microcontroller, or state machine. Aprocessor may also be implemented as a combination of computing devices,e.g., a combination of a DSP and a microprocessor, a plurality ofmicroprocessors, one or more microprocessors in conjunction with a DSPcore, or any other such configuration. Alternatively, some steps ormethods may be performed by circuitry that is specific to a givenfunction.

Additionally, one or more of the functions and operations describedherein may be performed by a processor, circuit, or other hardwareexecuting machine readable instructions stored in a non-transitorycomputer readable medium, which may comprise RAM (random access memory),ROM (read only memory), EPROM (erasable, programmable ROM), EEPROM(electrically erasable, programmable ROM), hard drives, flash memory, orother types of storage devices, which may be volatile and/ornonvolatile. It should be appreciated that other hardware componentsother than what is shown in any of the exemplary figures may also beprovided.

As described above, conventional in-band OSNR measurements have beenproposed that require foreknowledge of the modulation format andbit-rate of the optical signal, as well as careful calibration of theapparatus with a noiseless signal. In addition, sensitivity to signaldistortions introduced by CD and/or PMD may be a problem. Not only dotechniques described herein address these and other problems, but areliable maximum correlation when polarization-dependent loss (PDL) ispresent on the fiber link may also be provided.

FIG. 19 illustrates a flow chart of a method for mitigating polarizationdependent loss (PDL) in an optical signal-to-noise ratio (OSNR)measurement, according to an example. The method 1900 is provided by wayof example, as there may be a variety of ways to carry out the methoddescribed herein. The method 1900 may be executed or otherwise performedby one or more processing components as described herein, or by anothersystem or a combination of systems. Each block shown in FIG. 19 mayfurther represent one or more processes, methods, or subroutines, andone or more of the blocks may include machine readable instructionsstored on a non-transitory computer readable medium and executed by aprocessor or other type of processing circuit to perform one or moreoperations described herein.

At block 1910, an optical power spectrum of a modulated optical signalmay be measured. For example, a spectrum analyzer may measure theoptical power spectrum, where the optical power spectrum comprises atleast one of the plurality of wavelength channels.

At block 1920, portions of modulated optical signals may be selected.For example, a first portion of the modulated optical signal and asecond portion of the portions of the modulated optical signal may beselected by a measuring unit. In an example, each of the first andsecond portions of the optical portions of the modulated optical signalsmay comprise an independent noise distribution indicative ofpolarization dependent loss (PDL).

In some examples, the independent noise distribution may be introducedduring signal transmission randomly varies in 10 Hz cycles, andtransforming any elliptical polarization associated with the independentnoise distribution into a ball polarization is in 1 kHz speed.

At block 1930, a time-varying parameter of the first and second portionsof the modulated optical signal may be measured. For example, ameasuring unit may measure the time-varying parameter for the selectedportions.

At block 1940, a signal processor may determine a polarization dependentloss (PDL) mitigated optical signal-to-noise ratio (OSNR). For example,this may be achieved by identifying the independent noise distributionintroduced during signal transmission. Furthermore, any ellipticalpolarization associated with the independent noise distribution may betransformed into a ball polarization.

In some examples, transforming any elliptical polarization associatedwith the independent noise distribution into a ball polarization maycomprise determining a state of polarization (SOP). In some examples, acorrelation time interval may be used for an integration duration whendetermining the state of polarization (SOP).

In some examples, transforming any elliptical polarization associatedwith the independent noise distribution into a ball polarization mayalso involve applying Jones matrix multiplications to the first andsecond portions of the optical portions of the modulated optical signalsto yield an output. It may also involve applying Stokes vectorcalculations to the output. For example, the Stokes vector is appliedsuch that: S0 represents sum of power, S1 represents vertical/horizontalpower unbalance, S2 represents 45/−45 degree power unbalance, and S3represents right/left circular power unbalance. It should be appreciatedthat this may also include applying a feedback loop to get S2, S3 tozero. In some examples, the Stokes vector and feedback loop may becalculated for every integration interval. In some examples, the Jonesmatrix multiplications may applied in analog-to-digital converter (ADC)sample speed, and the Stokes vector and feedback loop may be calculatedfor every integration interval.

In addition, the signal processor may determine a correlation betweentime-varying parameters of the first and second portions of themodulated optical signal. Moreover, the signal processor may calculatethe polarization dependent loss (PDL) mitigated optical signal-to-noiseratio (OSNR) based at least in part on the correlation of thetime-varying parameters, the optical power spectrum of the modulatedoptical signal, and the ball polarization.

It should also be appreciated that the test instrument 200 may alsoprovide other components not shown. For example, middleware (not shown)may be included as well. The middleware may include software hosted byone or more servers or devices. Furthermore, it should be appreciatedthat some of the middleware or servers may or may not be needed toachieve functionality. Other types of servers, middleware, systems,platforms, and applications not shown may also be provided at theback-end to facilitate the features and functionalities of the testingand measurement system.

Moreover, single components may be provided as multiple components, andvice versa, to perform the functions and features described herein. Itshould be appreciated that the components of the system described hereinmay operate in partial or full capacity, or it may be removed entirely.It should also be appreciated that analytics and processing techniquesdescribed herein with respect to the test instrument 200, for example,may also be performed partially or in full by other various componentsof the overall system.

It should be appreciated that the data stores described herein mayinclude volatile and/or nonvolatile data storage that may store data andsoftware or firmware including machine-readable instructions. Thesoftware or firmware may include subroutines or applications thatperform the functions of the test instrument 200 and/or run one or moreapplication that utilize data from the test instrument 200 or othercommunicatively coupled system.

The various components, circuits, elements, and interfaces, may be anynumber of hardware, network, or software components, circuits, elements,and interfaces that serves to facilitate communication, exchange, andanalysis data between any number of or combination of equipment,protocol layers, or applications. For example, the interfaces describedherein may each include a network interface to communicate with otherservers, devices, components or network elements via a network.

Although examples are directed to measuring radio frequency (RF) powerin common public radio interface (CPRI) spectrum analysis of a cellsite, it should be appreciated that that the test instrument 200 mayalso use these and other various techniques in to provide interferenceanalysis, signal analysis, and/or other related RF measurements.Ultimately, the systems and methods described herein may minimize celltowner climbs and improve safety, reduce number of testing devices,increase accuracy and reliability, and significantly reduce maintenanceand operation expenses.

What has been described and illustrated herein are examples of thedisclosure along with some variations. The terms, descriptions, andfigures used herein are set forth by way of illustration only and arenot meant as limitations. Many variations are possible within the scopeof the disclosure, which is intended to be defined by the followingclaims—and their equivalents—in which all terms are meant in theirbroadest reasonable sense unless otherwise indicated.

The invention claimed is:
 1. An apparatus comprising: a signal processorto mitigate polarization dependent loss (PDL) in an opticalsignal-to-noise ratio (OSNR) by: measuring an optical power spectrum ofa modulated optical signal, wherein the optical power spectrum comprisesat least one of a plurality of wavelength channels; transforming anyelliptical polarization associated with independent noise distributioninto a ball polarization; determining a correlation between time-varyingparameters of first and second portions of the modulated optical signal,wherein each of the first and second portions of the modulated opticalsignal comprises an independent noise distribution indicative ofpolarization dependent loss (PDL); and calculating a polarizationdependent loss (PDL) mitigated optical signal-to-noise ratio (OSNR)based on the correlation between the time-varying parameters, theoptical power spectrum of the modulated optical signal, and the ballpolarization.
 2. The apparatus of claim 1, wherein the PDL introducedduring signal transmission randomly varies in 10 Hz cycles, andtransforms any elliptical polarization into a ball polarization in 1 kHzspeed.
 3. The apparatus of claim 1, wherein transforming any ellipticalpolarization associated with the independent noise distribution into aball polarization comprises determining a state of polarization (SOP).4. The apparatus of claim 3, wherein a correlation time interval is usedfor an integration duration when determining the state of polarization(SOP).
 5. The apparatus of claim 1, wherein transforming any ellipticalpolarization associated with the independent noise distribution into aball polarization comprises: applying Jones matrix multiplications tothe first and second portions of the modulated optical signal to yieldan output; applying Stokes vector calculations to the output, such that:S0 represents sum of power, S1 represents vertical/horizontal powerunbalance, S2 represents 45/−45 degree power unbalance, and S3represents right/left circular power unbalance; and applying a feedbackloop to get S1, S2, S3 to zero, wherein the Stokes vector calculationsand feedback loop are calculated for every integration interval.
 6. Theapparatus of claim 5, wherein the Jones matrix multiplications areapplied in analog-to-digital converter (ADC) sample speed, and whereinthe Stokes vector calculations and feedback loop are calculated forevery integration interval.
 7. A method comprising: measuring an opticalpower spectrum of a modulated optical signal, wherein the optical powerspectrum comprises at least one of a plurality of wavelength channels;selecting a first portion of the modulated optical signal and a secondportion of the modulated optical signal, wherein each of the first andsecond portions of the modulated optical signal comprises an independentnoise distribution indicative of polarization dependent loss (PDL);transforming any elliptical polarizations associated with theindependent noise distribution into a ball polarization; determining acorrelation between time-varying parameters of the first and secondportions of the modulated optical signal; and determining a polarizationdependent loss (PDL) mitigated optical signal-to-noise ratio (OSNR)based on the correlation between the time-varying parameters, theoptical power spectrum of the modulated optical signal, and the ballpolarization.
 8. The method of claim 7, further comprising: measuringthe time-varying parameters of the first and second portions of themodulated optical signal.
 9. The method of claim 8, wherein theindependent noise distribution introduced during signal transmissionrandomly varies in 10 Hz cycles, and transforming any ellipticalpolarization associated with the independent noise distribution into aball polarization is in 1 kHz speed.
 10. The method of claim 8, whereintransforming any elliptical polarization associated with the independentnoise distribution into a ball polarization comprises determining astate of polarization (SOP).
 11. The method of claim 10, wherein acorrelation time interval is used for an integration duration whendetermining the state of polarization (SOP).
 12. The method of claim 8,wherein transforming any elliptical polarization associated with theindependent noise distribution into a ball polarization comprises:applying Jones matrix multiplications to the first and second portionsof the modulated optical signal to yield an output; and applying Stokesvector calculations to the output.
 13. The method of claim 12, wherein:the Stokes vector is applied such that: S0 represents sum of power, S1represents vertical/horizontal power unbalance, S2 represents 45/−45degree power unbalance, and S3 represents right/left circular powerunbalance; and applying a feedback loop to get S1, S2, S3 to zero,wherein the Stokes vector calculations and feedback loop are calculatedfor every integration interval.
 14. The method of claim 13, wherein theJones matrix multiplications are applied in analog-to-digital converter(ADC) sample speed.
 15. The method of claim 13, wherein the Stokesvector calculations and feedback loop are calculated for everyintegration interval.
 16. A non-transitory computer-readable storagemedium having an executable stored thereon, which when executedinstructs a processor to perform the method of claim
 7. 17. A methodcomprising: measuring an optical power spectrum of a modulated opticalsignal, wherein the optical power spectrum comprises at least one of aplurality of wavelength channels: transforming any ellipticalpolarization associated with independent noise distribution into a ballpolarization; determining a correlation between time-varying parametersof first and second portions of the modulated optical signal, whereineach of the first and second portions of the modulated optical signalcomprises an independent noise distribution indicative of polarizationdependent loss (PDL); and calculating a polarization dependent loss(PDL) mitigated optical signal-to-noise ratio (OSNR) based on thecorrelation between the time-varying parameters, the optical powerspectrum of the modulated optical signal, and the ball polarization. 18.The method of claim 17, wherein transforming any elliptical polarizationassociated with the independent noise distribution into a ballpolarization comprises at least one of: determining a state ofpolarization (SOP), and wherein a correlation time interval is used foran integration duration when determining the state of polarization(SOP); applying Jones matrix multiplications to the first and secondportions of the modulated optical signal to yield an output; or applyingStokes vector calculations to the output such that: S0 represents sum ofpower, S1 represents vertical/horizontal power unbalance, S2 represents45/−45 degree power unbalance, and S3 represents right/left circularpower unbalance; and applying a feedback loop to get S1, S2, S3 to zero,wherein the Stokes vector calculations and feedback loop are calculatedfor every integration interval.
 19. The method of claim 18, wherein theJones matrix multiplications are applied in analog-to-digital converter(ADC) sample speed, and wherein the Stokes vector calculations andfeedback loop are calculated for every integration interval.
 20. Anon-transitory computer-readable storage medium having an executablestored thereon, which when executed instructs a processor to perform themethod of claim 17.